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Chapter 19: Problem 44

A straight wire segment of length \(0.60 \mathrm{m}\) carries a current of $18.0 \mathrm{A}$ and is immersed in a uniform external magnetic ficld of magnitude \(0.20 \mathrm{T}\). (a) What is the magnitude of the maximum possible magnetic force on the wire segment? (b) Explain why the given information enables you to calculate only the maximum possible force.

Short Answer

Expert verified
Answer: The maximum magnetic force on the wire segment is 2.16 N. Only the maximum possible force can be calculated because we do not have the information about the angle between the current and the magnetic field. However, since we know the maximum value of sinθ is 1 (when θ = 90°), we can calculate the maximum possible force considering the angle to be 90°. Any other angle would result in a lesser force, but we do not have enough information to find the exact force.

Step by step solution

01

Identify given variables

We are given the following values: - Length of the wire (L) = \(0.60 m\) - Current in the wire (I) = \(18.0 A\) - Magnitude of the uniform external magnetic field (B) = \(0.20 T\)
02

Calculate the maximum force

We can calculate the magnetic force on the wire segment using the formula: $$ F = ILB \sin\theta $$ To find the maximum value for the force, observe that \(\sin\theta\) can take values between -1 and 1, with the maximum value being \(\sin {90^\circ} =1\). So, to find the maximum force, set \(\theta = {90^\circ}\): $$ F_{max} = ILB\sin{90^\circ} = ILB $$ Now, plug in the given values and compute F_max: $$ F_{max} = (18.0\,\mathrm{A})(0.60\,\mathrm{m})(0.20\,\mathrm{T}) = 2.16\,\mathrm{N} $$ The maximum possible magnetic force on the wire segment is \(2.16\,\mathrm{N}\).
03

Explain why only the maximum possible force can be calculated

The given information does not provide the angle between the current and the magnetic field, which means the actual force could be lower than the maximum possible force, depending on the value of \(\theta\). However, since we know the maximum value of \(\sin\theta\) is 1, we can calculate the maximum possible force when \(\theta=90^\circ\). Any other angle between current and magnetic field would result in a lesser force. And thus, only the maximum possible force can be calculated with the given information.

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